The square root of a number n is the value that, multiplied by itself, gives n — written √n. Every positive number has two real square roots, +√n and −√n, though the symbol √ refers to the positive (principal) one. When n is a perfect square the root is a whole number (√16 = 4); otherwise it's an irrational decimal (√2 ≈ 1.414). This calculator gives the principal root, both roots, the cube root, and a perfect-square check.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: standard root definitions, recomputed in code.
Roots, defined
Squaring and taking a square root are inverse operations, so √(x²) = |x|. Because two different numbers square to the same value, every positive n has both a positive and a negative root. A perfect square is simply a number whose principal root is a whole number. The cube root works the same way but with three equal factors — and unlike the square root, it accepts negative inputs.
Worked example — 16, 2 and 50
Scenario: one perfect square and two that aren't.
√16 is exactly 4 because 16 is a perfect square. √2 is 1.414214, an irrational number whose decimals never repeat. √50 isn't a perfect square, but it simplifies to 5√2 ≈ 7.071068 by pulling out the perfect-square factor 25. For comparison the cube root of 27 is exactly 3 (3³ = 27), while √27 ≈ 5.196.
Frequently Asked Questions
It's the value that times itself gives n. √16 = 4 (4×4=16). Non-perfect squares give irrational decimals like √2 = 1.414214.
Yes — a positive and a negative, since (+x)² = (−x)². Roots of 16 are +4 and −4. The √ symbol means the positive one.
A whole number with a whole-number root: 1, 4, 9, 16, 25… 81 = 9² and 1,000,000 = 1000². Others (2, 50, 27) are irrational.
Not in real numbers — √(−4) is imaginary (2i). This tool needs n ≥ 0 for the square root. Cube roots of negatives are fine.
Square root undoes squaring; cube root undoes cubing. √27 ≈ 5.196 but ∛27 = 3 (3³ = 27). ∛(−8) = −2.